Lemma: Kinematical nilpotence

The rule (eq:nonlinear) is nilpotent on the off-shell graded field algebra.

Proof. Let $w^{\mu }=c^{\nu }{\partial }_{\nu }{c}^{\mu }$. The graded Lie-derivative identity is ${{\mathcal{L}}_c}^2={\mathcal{L}}_w$. It follows directly for scalars and covectors by expansion; the symmetric second-derivative term vanishes by anticommutativity of the ghosts. The identity extends to tensor products by the graded Leibniz rule.

Since $sq=\kappa {\mathcal{L}}_{c}q$,

The scalar calculation is identical. Direct expansion of $s(c^{\nu }{\partial }_{\nu }{c}^{\mu })$ gives zero after substitution of $sc$ and antisymmetrization. The nonminimal pair is nilpotent by definition.